Modeling fish migration with an interacting particle model

Alethea Barbaro

CWRUDepartment of Mathematics, Applied Mathematics & Statistics

17 April 2015

A. Barbaro (CWRU) Modeling fish migration with an interacting particle model 17 April 2015 1 / 43

Research summary related to interacting particle models:

Interacting particle model for fish migrationBehavior of the particle model with noise (with B. Birnir, K. Taylor; simulation assistance from P.Trethewey, L. Youseff)Parallelization of the simulations (with B. Birnir, J. Gilbert, P. Trethewey, L. Youseff)The model applied to the Icelandic Capelin (with B. Birnir, B. Einarsson, S. Sigurdsson, The MarineInstitute of Iceland)Scaling in interacting particle systems (with B. Einarsson) [current]

Interacting particle models for gang dynamicsA coupled network model for gang rivalry formation (with R. Hegemann, L. Smith, S. Reid, A.Bertozzi, G. Tita)A statistical mechanics approach to gang territorial development (with L. Chayes, M. R. D’Orsogna)

Kinetic and hydrodynamic models for particle systemsPhase transition and diffusion among socially interacting self-propelled agents (with P. Degond)Phase transition in a kinetic Cucker-Smale model with self-propulsion and friction (with J. A. Carrillo,P. Degond) [current]A kinetic contagion model for fear in crowds (with J. Rosado) [current]An exploration of the effect of normalization and different kinds of noise in Vicsek-type flockingmodels (with M. Burger) [current]

The Data: Icelandic stock of capelin

The Icelandic stock of capelin

An example of the acoustic data:

Our model

(xk (t + ∆t)yk (t + ∆t)

(xk (t)yk (t)

)+ ∆t · vk (t)

Dk (t)‖ Dk (t) ‖

+ C(p̃k (t))

Here, Dk is the directional heading of particle k

∆t is the timestep

Particle k ’s position in the plane is pk

p̃k is the nearest gridpoint to pk

C(p̃k ) is the current at p̃k

Choosing the direction

The directional heading of the particles (apart from environmental effects) is determined as follows:(cos(φk (t + ∆t))sin(φk (t + ∆t))

dk (t + ∆t)‖ dk (t + ∆t) ‖

dk (t + ∆t) :=

( ∑r∈Rk

pk (t)− pr (t)‖ pk (t)− pr (t) ‖

o∈Ok

(cos(φo(t))sin(φo(t))

a∈Ak

pa(t)− pk (t)‖ pa(t)− pk (t) ‖

radius of orientation

Zone of Attraction

Zone of Orientation

Zone of Repulsion

radius o

f attra

radius of repulsion

Environmental information

Febrúar 1985 Hámarkshitastig: 7.16°C

−30 −25 −20 −15 −1062

Febrúar 1991 Hámarkshitastig: 7.47°C

−30 −25 −20 −15 −1062

11. febrúar 2008 Hámarkshitastig: 8.99°C

−30 −25 −20 −15 −1062

February, 1985 February 1991 February 2008 1

1http://www.wetterzentrale.de/topkarten/fsfaxsem.html

Temperature function

Function r(T ) determines the reaction of the particles to the temperature field.

r(T ) :=

−(T − T1)4 if T ≤ T10 if T1 ≤ T ≤ T2

−(T − T2)2 if T2 ≤ T

The model

(xk (t + ∆t)yk (t + ∆t)

(xk (t)yk (t)

)+ ∆t · vk (t)

Dk (t)‖ Dk (t) ‖

+ C(p̃k (t))

Dk (t + ∆t) :=

cos(φk (t + ∆t))sin(φk (t + ∆t))

)︸︷︷︸

interaction term

+β∇r(T (pk (t))

)‖ ∇r

(T (pk (t))

)‖︸︷︷︸

temperature term

for α+ β = 1.

Acoustic data from 1984-1985

(a) (b)

Figure : The distribution of capelin during the spawning migration of 1984-1985.(a) Acoustic data from November 1 to November 21 (b) Acoustic data from January 14 to February 8(c) Close up of the distribution of capelin from February 7 to February 20 of 1985.

Acoustic data from 1990-1991

(a) (b)

Figure : The distribution of capelin during the spawning migration of 1991.(a) Acoustic data from January 4 to January 11.(b) Close up of the distribution of capelin southeast of Iceland from February 8 to February 9 of 1991.(c) Acoustic data from February 17 to February 18.

1984-1985

1990-1991

(a) (b)

(c) (d)

Figure : Simulation of the 2007-2008 spawning migration.(a) Early January, day 0(b) Mid-February, day 47(c) Late February, day 59(d) Early March, day 65.

Acoustic data from 2008

Figure : Collected migration data:(a) Measured distribution of capelin near south coast of Iceland from February 26 to February 27 of 2008.(b) Measured distribution of capelin near the southeast coast of Iceland from February 29 to March 3 of 2008.

Sensitivity to perturbed parameters

We measure the sensitivity of the system by seeing how the migration route and timing change.For details, see to [2].

The problem of superindividuals

In the real migrations which we are trying to accurately capture, it is safe to assume there arearound 5 · 1010 fish

In our simulations, we use roughly 5 · 104 particles

This means each particle represents 106 fish

Each particle must therefore be thought of as a superindividual

With these superindividuals, we captured the migration

The goal

One fish per particle

Then we could more confidently justify our behavioral rules, since they are based on dataobtained from interactions among individual fish

So this leads to a question: how does the system change as we change the number of particles?

We need to make some assumptions:

We assume uniform density of particles and fish in the schools

The interaction length of the particles should be much less than the size of the school

We further assume the velocities of the particles are equal

The goal

One fish per particle

Then we could more confidently justify our behavioral rules, since they are based on dataobtained from interactions among individual fish

So this leads to a question: how does the system change as we change the number of particles?We need to make some assumptions:

We assume uniform density of particles and fish in the schools

The interaction length of the particles should be much less than the size of the school

We further assume the velocities of the particles are equal

Propagation of information through the school

If sufficiently dense, local interactions between particles allows information to propagatethrough a school

Temperature informationInformation about predatorsInformation about food

We want to preserve the speed at which this information propagates through the school

Varying Numbers of Particles

Relationship between Parameters in the Simulation

In the actual migration, there are a given number of fish within a given area

Each simulation needs to relate back to this real situation:

# Real fish

Size of domain=

(# Real fish

# Particles in simulation

)(# Particles in simulation

# Zones of interaction

)(# Zones of interaction

Size of domain

)= (Fish per particle)

(# Particles in simulation

Size of domain

D/πRi2

)(D/πRi

When particles are uniformly distributed, the second term is roughly the number of interactionneighbors per particle, which is close to uniform in space. Calling this Mi gives:

# Real fishSize of domain =

)(Mi )

# Real fish

Size of domain=

(# Real fish

Size of domain

D/πRi2

)(D/πRi

)(Mi )

# Real fish

Size of domain=

(# Real fish

Size of domain

D/πRi2

)(D/πRi

)(Mi )

# Real fish

Size of domain=

(# Real fish

Size of domain

D/πRi2

)(D/πRi

)(Mi )

Index the simulation where we captured the migration by 0.

Index a new simulation by 1.# Real fish

Size of domain remains constant, so:

)⇒ 1

Consider number of interaction partners to be fixed. Then:

R20 N0N1⇒ R1 = R0

1√N1

So, if we want to maintain the number of interaction partners, the radii and the number ofparticles should relate as follows:

R ∝ 1√N.

)⇒ 1

R20 N0N1⇒ R1 = R0

1√N1

R ∝ 1√N.

)⇒ 1

R20 N0N1⇒ R1 = R0

1√N1

R ∝ 1√N.

)⇒ 1

R20 N0N1⇒ R1 = R0

1√N1

R ∝ 1√N.

)⇒ 1

R20 N0N1⇒ R1 = R0

1√N1

R ∝ 1√N.

In discrete system, want the portion of the zone traversed per timestep to remain constant aswe vary the number of particles

So that a particle does not pass outside its zone in one timestep

To guarantee this, v∆t = cR where v and c are constant as we vary the number of particles

In this way, we see:∆t ∝ R ∝ 1√

Constant density of actual fish

Each simulation needs to relate back to this real situationSchematic:

fishregion = ( particles

interaction·zone )( fishparticle )( interaction·zone

region )

Let N denote the total number of particles in a simulation, F denote the number of fish in themigration, and Aw denote the total area of the regionLet M denote the number of particles per interaction zone

Constant across interaction zones due to constant density assumptionFor computational intensity, need M is constant across different simulations (so the number ofneighbors for each particle remains constant)

region )

Let N denote the total number of particles in a simulation, F denote the number of fish in themigration, and Aw denote the total area of the region

Let M denote the number of particles per interaction zoneConstant across interaction zones due to constant density assumptionFor computational intensity, need M is constant across different simulations (so the number ofneighbors for each particle remains constant)

region )

Let N denote the total number of particles in a simulation, F denote the number of fish in themigration, and Aw denote the total area of the regionLet M denote the number of particles per interaction zone

Constant across interaction zones due to constant density assumptionFor computational intensity, need M is constant across different simulations (so the number ofneighbors for each particle remains constant)

Relating time and space to the number of particles

Then for a given simulation indexed by i , FAw

= (M)( FNi

)( Awπr2

i)⇒ 1

= M(πr2

For two different simulations:M

(πr20 )N0

= M(πr2

1 )N1⇒ (

)2 =N0N1

r1 = r0

√N0N1

Considering r0 and N0 to have come from a reference simulation:

∆t ∝ r ∝√

= (M)( FNi

)( Awπr2

i)⇒ 1

= M(πr2

(πr20 )N0

= M(πr2

1 )N1⇒ (

)2 =N0N1

r1 = r0

√N0N1

∆t ∝ r ∝√

= (M)( FNi

)( Awπr2

i)⇒ 1

= M(πr2

(πr20 )N0

= M(πr2

1 )N1⇒ (

)2 =N0N1

r1 = r0

√N0N1

∆t ∝ r ∝√

= (M)( FNi

)( Awπr2

i)⇒ 1

= M(πr2

(πr20 )N0

= M(πr2

1 )N1⇒ (

)2 =N0N1

r1 = r0

√N0N1

∆t ∝ r ∝√

Our parameters for the migrations

∆t = 0.05 days

Initial speed vk ' 4− 8 km/day

rr = 0.01 or about ∼ 120 m

ro = ra = 0.1 or about ∼ 1.2 km

Number of particles is roughly 5 · 104

Scaling down to an individual level

How do the particles scale as we take Ns to 1? A rough estimate for the total number of fish in amigration is F ' 5 · 1010.

N0 ' 5 · 104 and N1 ' 5 · 1010

∆t0 = 0.05 days and ∆t0∆q0

= ∆t1∆q1⇒ ∆t1 = 4.32 seconds

∆q01√N0

= ∆q11√N1

and ∆q0 ' 1.2 km⇒ ∆q1 ' 1.2 meters

Radii scale with ∆q, sorr0 ' 120 meters⇒ rr1 ' 12cmro0 = ra0 ' 1.2 km⇒ ro1 = ra1 ' 1.2m

These are all biologically reasonable!

N0 ' 5 · 104 and N1 ' 5 · 1010

∆t0 = 0.05 days and ∆t0∆q0

= ∆t1∆q1⇒ ∆t1 = 4.32 seconds

∆q01√N0

= ∆q11√N1

and ∆q0 ' 1.2 km⇒ ∆q1 ' 1.2 meters

N0 ' 5 · 104 and N1 ' 5 · 1010

∆t0 = 0.05 days and ∆t0∆q0

= ∆t1∆q1⇒ ∆t1 = 4.32 seconds

∆q01√N0

= ∆q11√N1

and ∆q0 ' 1.2 km⇒ ∆q1 ' 1.2 meters

N0 ' 5 · 104 and N1 ' 5 · 1010

∆t0 = 0.05 days and ∆t0∆q0

= ∆t1∆q1⇒ ∆t1 = 4.32 seconds

∆q01√N0

= ∆q11√N1

and ∆q0 ' 1.2 km⇒ ∆q1 ' 1.2 meters

N0 ' 5 · 104 and N1 ' 5 · 1010

∆t0 = 0.05 days and ∆t0∆q0

= ∆t1∆q1⇒ ∆t1 = 4.32 seconds

∆q01√N0

= ∆q11√N1

and ∆q0 ' 1.2 km⇒ ∆q1 ' 1.2 meters

N0 ' 5 · 104 and N1 ' 5 · 1010

∆t0 = 0.05 days and ∆t0∆q0

= ∆t1∆q1⇒ ∆t1 = 4.32 seconds

∆q01√N0

= ∆q11√N1

and ∆q0 ' 1.2 km⇒ ∆q1 ' 1.2 meters

Where to go from here

Toward DataEinarsson, Birnir, and Sigurdsson have created a dynamic energy budget (DEB) model for thephysiology of the capelin [9]

Next step: Incorporate this DEB model into the simulations of the spawning migration

Toward Mathematics

Numerical validation of the proposed scaling laws

Kinetic and hydrodynamic versions of similar models have been and are being studied

Models taking into consideration the number of interaction neighbors have also beenproposed and studied

Including emotional influences into the model

Where to go from here

Toward DataEinarsson, Birnir, and Sigurdsson have created a dynamic energy budget (DEB) model for thephysiology of the capelin [9]

Next step: Incorporate this DEB model into the simulations of the spawning migration

Toward MathematicsNumerical validation of the proposed scaling laws

Kinetic and hydrodynamic versions of similar models have been and are being studied

Models taking into consideration the number of interaction neighbors have also beenproposed and studied

Including emotional influences into the model

Gangs are a problem in Los Angeles!

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Violence data: 1998, 1999, and 2000

The Rivalry Network

Observed Rivalry Network Among Hollenbeck Gangs 2

29 Active Gangs in Hollenbeck

69 Rivalries Among the Gangs

A Set Space is a gang’s center of activity where gangmembers spend a large quantity of their time

Gang activity in Hollenbeck is generally isolated from gangactivity outside of Hollenbeck

Freeways and other geographic features influence the rivalrynetwork

2S. Radil, C. Flint, and G. Tita,“Spatializing Social Networks: Using Social Network Analysis to Investigate Geographies of Gang Rivalry, Territoriality, andViolence in Los Angeles.” 2010.

A control: just diffusion

What’s going wrong?

Potential models

Graph Generating MethodsGeographical Threshold Graphs

Agent-Based MethodsBrownian Motion with Semi-Permeable BoundariesBiased Lévy Flights with Semi-Permeable Boundaries

Coupling the rivalry network and avoidance strengthDecay on the edges of graphHeading homeAvoiding rivals’ set spacesSemi-permeable freeways

Potential models

Geographical Threshold Graphs 3

Geographical Threshold Graphs (GTGs) randomly assign weights ηi to the N nodes

The edge between nodes ni and nj exists only ifF (ηi ,ηj )

d(ni ,nj )β≥ Threshold

We construct a specific realization of GTGs:

ηi = size of gang i

F (ηi , ηj ) = ηi · ηj , and β = 2

Threshold to have the same number of rivalries as observed network

3M. Bradonjic, A. Hagberg, A. Percus. Giant Component and Connectivity in Geographical Threshold Graphs (2007).

Biased Lévy walk with semi-permeable boundaries

Movement dynamics:

Agents move in free space according to a biased Lévy walk

Choose direction of bias according to location of other gangs’ set spaces and location of the agent’s ownset space

Agents have some probability of crossing a boundary

Interactions:

If two gang members from different gangs cross paths, then an interaction has occurred and the rivalrybetween the gangs is excited

At the end of the simulation, we exclude rivalries where the number of interactions is mutually insignificantto both gangs

Comparison of Networks: Observed, Geogrpahical Threshold Graph (GTG), Brownian Motion Network (BMN), Simulated

Biased Lévy walk Network (SBLN)

Observed GTG BMN SBLN

Limiting Behavior of ensemble SBLN: graph density

Density Variance CentralityObserved 0.169951 4.32105 0.201058

GTG 0.169951 9.976219 0.277778Ensemble 0.163547 3.6642331 0.1503968

BSN ± 0.005593 ± 0.483954 ± 0.018831

Table : The table provides the shape measures for the observed network, GTG, BMN, and ensemble BSN.

Performance of the models

SBLN and GTG both performed quite well in metric comparisons (accuracy, shape, community structuremetrics)

SBLN allows us to explore evolution of the rivalries

SBLN produces dynamic stochastic networks:

Comparison (left to right) of ensemble SBLN 100% edge agreement, ensemble SBLN 50% edge agreement, and ensemble SBLN 1% edge

agreement

Static network model vs. SBLN model

SBLN allows us to see where interactions take place

Simulated 1998 1999 2000

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M. R. D’Orsogna, Y. L. Chuang, A. L. Bertozzi and L. Chayes, Self-propelled particles withsoft-core interactions: patterns, stability and collapse, Phys. Rev. Lett., 96 (2006), 104302.

B. Einarsson, B. Birnir, S. Sigurdsson, A dynamic energy budget (DEB) model for the energyusage and reproduction of the Icelandic capelin (Mallotus villosus), Journal of theoreticalbiology, 281(1), (2011), 1–8.

C. K. Hemelrijk and H. Hildenbrandt, Some causes of the variable shape of flocks of birds,PLOS ONE, 6 (2011), e22479.

T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen and O. Shochet, Novel type of phase transitionin a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226–1229.